The sum of three positive numbers is 18 and their product is 162. If the sum of two numbers is equal to the third number, then the sum of squares of the numbers is

Options

A. A) 120
B. B) 126
C. C) 132
D. D) 138

Answer

To solve the problem, let's denote the three positive numbers as \( A \), \( B \), and \( C \). We are given the following information:

1. The sum of the three numbers is 18:
\[ A + B + C = 18 \]

2. The product of the three numbers is 162:
\[ A \cdot B \cdot C = 162 \]

3. The sum of two numbers is equal to the third number:
\[ A + B = C \]

### Step 1: Substitute \( C \) in the sum equation
From the third equation, we can express \( C \) in terms of \( A \) and \( B \):
\[ C = A + B \]
Now, substitute this expression for \( C \) into the first equation:
\[ A + B + (A + B) = 18 \]
This simplifies to:
\[ 2(A + B) = 18 \]
Dividing both sides by 2 gives:
\[ A + B = 9 \]

### Step 2: Substitute \( A + B \) in the product equation
Now, we can substitute \( C \) in the product equation using \( C = A + B = 9 \):
\[ A \cdot B \cdot 9 = 162 \]
Dividing both sides by 9 gives:
\[ A \cdot B = \frac{162}{9} = 18 \]

### Step 3: Solve for \( A \) and \( B \)
Now we have two equations:
1. \( A + B = 9 \)
2. \( A \cdot B = 18 \)

Let \( A \) and \( B \) be the roots of the quadratic equation \( x^2 - (A+B)x + AB = 0 \):
\[ x^2 - 9x + 18 = 0 \]

### Step 4: Factor the quadratic equation
To factor the quadratic equation:
\[ x^2 - 9x + 18 = (x - 6)(x - 3) = 0 \]
Thus, the roots are:
\[ A = 6, \quad B = 3 \]

### Step 5: Find \( C \)
Using \( A + B = C \):
\[ C = 6 + 3 = 9 \]

### Step 6: Calculate the sum of squares
Now we need to find the sum of the squares of the numbers:
\[ A^2 + B^2 + C^2 = 6^2 + 3^2 + 9^2 \]
Calculating each square:
\[ 6^2 = 36, \quad 3^2 = 9, \quad 9^2 = 81 \]
Now, summing these values:
\[ A^2 + B^2 + C^2 = 36 + 9 + 81 = 126 \]

### Final Answer
The sum of the squares of the numbers is:
\[ \boxed{126} \]